A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic 1st Edition by Jeffrey Bergen (PDF)

Description

A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. The text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics wich arise in courses in algebra, geometry, trigonometry, precalculus and calculus. The final four chapters present the more theoretical material needed for graduate study.Presents a more natural ‘rings first’ approach to effectively leading the student into the the abstract material of the course by the use of motivating concepts from previous math courses to guide the discussion of abstract algebraBridges the gap for students by showing how most of the concepts within an abstract algebra course are actually tools used to solve difficult, but well-known problems Builds on relatively familiar material (Integers, polynomials) and moves onto more abstract topics, while providing a historical approach of introducing groups first as automorphisms Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is an amazing abstract algebra book for self-study. Unlike most abstract algebra books that have a boring theorem/proof/corollary approach, this is a very “chatty” text that clocks in at almost 700 pages. I find that many people who want to become mathematicians do not like math books that have pages of text, but for those of us who really have trouble pressing forward if we do not understand the “why” of the math, this type of book is excellent. Whenever a concept was slightly fuzzy, the author promised that it would be revisited and clarified in another section. I found that this was always the case.The text holds good on its promise of a concrete approach. Although the text presents the abstract ideas, it relates them to concrete examples that the student is already familiar with. The word concrete is used somewhat loosely because it often includes complex numbers, which the texts explains beautifully. All of the major ideas in the book are presented in terms of their historical motivation or how they are used to solve problems from previous math courses such as algebra, trig, geometry, and calculus. Some people may appreciate math just for its beauty, but I usually need to understand why I should care. That reason doesn’t have to be a physical application as long as I understand where the material is taking me or how it applies to topics I am familiar with. This text provides the reader with several interesting motivations and, unlike most abstract algebra texts, there are applications to trigonometry and calculus. One that really stands out is the proof that partial fraction decomposition is always possible for a rational function. The text also explains why a polynomial f(x) has a double root if and only if it shares that root with its derivative. Some basic results from calculus/analysis are also used to prove the fundamental theorem of algebra. By the time you get to the end of the book, you are well-equipped to understand the proof of the insolvability of the quintic and the impossibility of constructing a 20 degree angle. Because of all of the interesting motivations, someone who is not pursuing further study in math can benefit from this book. In contrast, most algebra books introduce terminology such as group, homomorphism, isomorphism, ideal, splitting field, quotient group, etc, without giving any explanation for why these concepts were created and how they are used.If you want to pursue graduate study in math, this book is probably not comprehensive enough for you, although you will learn a ton and will have a better idea of why the subject is important than if you read a more standard text. I read it purely for enjoyment, and now I feel prepared to tackle more advanced texts. The text is suitable for anyone who has made it through calculus II, but even someone who hasn’t had calculus can understand almost all of the material. Calculus is really only needed for some of the applications. You do need to understand some set notation, such as something like S={b ∈ N | there exists some integer a such that q=b/a}. This is the only time the book doesn’t totally hold your hand, but how to read that type of notation can easily be found elsewhere. All of the proofs are explained very well – you may have to read them more than once but that is true no matter how well a proof is written. They are definitely more readable than any other abstract algebra book I have found, and I have looked through several of the most popular texts.In a word, I think this is essential reading before you take an abstract algebra class that uses a more traditional text like Hungerford or Gallian.Another fantastic math book I would recommend is Number Theory: A Lively Introduction with Proofs, Applications, and Stories.

⭐This book is not good for elementary learners. The contexts contain lots of mathematical languages and general forms of theorems are presented which make the book more difficult to understand. It suits only for those who already have a high degree of knowledge of abstract algebra to read.

⭐Perfect great condition

⭐Some mathematical formulas are unreadable in kindle 3 and kindle dxg as they are generated in image formats. I also tried the kindle app in my android tablet (Xoom). The image formulas can be displayed in a very good way but some special notations cannot be showed properly (the notations for natural number set, interger set, complex number set, etc).Even though the content is really inspiring for self study, I will recommend to buy a paper book instead. The kindle version is poorly generated.

⭐This book is extremely good, especially if you are not an expert in abstract algebra beforehand. It is, contrary to what other reviewers said, quite good in explaining the subject. Everything is proved very thoroughly, no “the proof of this theorem is beyond the scope of this book” here. The book explains not only the “usual” parts of abstract algebra (groups, rings, fields etc), but also the fundamental theorem of algebra (which is often quoted, but not proved, in introductions to abstract algebra), the motivation for normal subgroups (very rarely motivated in other books) and how to find a formula for a given series (formulas which are often, in other books, proved by mathematical induction, but never explained as to how one can discover those formulas). The exercises are plenty, but none are hard. This book brings out the beauty of abstract algebra and as easy as it can possibly be done. I give this book my warmest recommendations for anyone who truly wants to understand abstract algebra up to Galois theory, from scratch.

⭐Downloaded trial after reading other user’s review on the text. Kindle version did not translate the character set correctly. Missing key symbols in equations. Makes exercises difficult to decipher at best.Kindle version should be removed as an option, but trial could get you by until paper version arrives in the mail.

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